## At A Glance

### UnivHypGeom16: Midpoints and bisectors

Midpoints of sides may be defined in terms of reflections in points in hyperbolic geometry. Reflections are defined by 2x2 trace zero matrices associated to points. The case of a reflection in a null point is somewhat special. The crucial property of reflection is that it preserves perpendicularity, which then implies that reflections send lines to lines. Midpoints of a side bc can be constructed with a straightedge when they exist, and in general there are two of them! This is a big difference with Euclidean geometry. Bisectors of vertices are defined by duality.CONTENT SUMMARY: pg 1: @00:10 point/matrix correspondence; reflection matrix conjugation theorem; exercise 16.1pg 2: @03:28 Definition of reflection of a general pointpg 3: @07:32 another example; Null reflection theorem; proof (exercise 16-2)pg? 4: @09:29 Matrix perpendicularity theorem; reflections as generators of isometries in hyperbolic geometrypg 5: @14:30 Reflection (preserves) perpendicularity theorem; remark about trace; proofpg 6: @18:11 reflection (preserves) lines theorem; proof; Line/point reflection notationpg 7: @21:24 exercise 16-3; Concept of Midpoint between 2 pointspg 8: @24:26 Geometrical construction concerning midpointspg 9: @28:59 Another geometrical construction concerning midpoints; Harmonic quadrangle and harmonic conjugates UHG2 revisitedpg 10: @31:42 another midpoints constructionpg 11: @33:34 Not? all sides have midpoints; side/vertex midpoints/bisectors (THANKS to EmptySpaceEnterprise)
Length: 36:41

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